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| Content Provider | Springer Nature Link |
|---|---|
| Author | Grunau, Hans Christoph |
| Copyright Year | 2009 |
| Abstract | The linear clamped plate boundary value problem is a classical model in mechanics. The underlying differential equation is elliptic and of fourth order. The latter is a peculiar feature with respect to which this equation differs from numerous equations in physics and engineering which are of second order. Concerning the clamped plate boundary value problem, “linear questions” may be considered as well understood. This changes completely as soon as one poses the simplest “nonlinear question”: What can be said about positivity preserving? Does a plate bend upwards when being pushed upwards? It is known that the answer is “no” in general. However, there are many positivity issues as e.g. “almost positivity” to be discussed.Boundary value problems for the “Willmore equation” are nonlinear counterparts for the linear clamped plate equation. The corresponding energy functional involves curvature integrals over the unknown surface. The Willmore equation is of interest in mechanics, membrane physics and, in particular, in differential geometry. Quite far reaching results were achieved concerning closed surfaces. As for boundary value problems, by far less is known. These will be discussed in symmetric situations.This survey article reports upon joint works with A. Dall’Acqua, K. Deckelnick (Magdeburg), S. Fröhlich (Free University of Berlin), F. Gazzola (Milan), F. Robert (Nice), Friedhelm Schieweck (Magdeburg) and G. Sweers (Cologne). |
| Ending Page | 204 |
| Page Count | 34 |
| Starting Page | 171 |
| File Format | |
| ISSN | 14249286 |
| e-ISSN | 14249294 |
| Journal | Milan Journal of Mathematics |
| Issue Number | 1 |
| Volume Number | 77 |
| Language | English |
| Publisher | SP Birkhäuser Verlag Basel |
| Publisher Date | 2009-08-08 |
| Publisher Place | Basel |
| Access Restriction | One Nation One Subscription (ONOS) |
| Subject Keyword | Maximum principles Boundary value problems for higher-order elliptic equations Immersions (minimal, prescribed curvature, tight, etc.) Positivity surfaces of revolution Nonlinear boundary value problems for linear elliptic equations Mathematics Willmore equation clamped plate equation Nonlinear ordinary differential operators Analysis Dirichlet problem Optimization of shapes other than minimal surfaces |
| Content Type | Text |
| Resource Type | Article |
| Subject | Mathematics |
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